Problem: Simplify and expand the following expression: $ \dfrac{5}{p + 6}- \dfrac{5}{4p + 20}- \dfrac{4}{p^2 + 11p + 30} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the second term: $ \dfrac{5}{4p + 20} = \dfrac{5}{4(p + 5)}$ We can factor the quadratic in the third term: $ \dfrac{4}{p^2 + 11p + 30} = \dfrac{4}{(p + 6)(p + 5)}$ Now we have: $ \dfrac{5}{p + 6}- \dfrac{5}{4(p + 5)}- \dfrac{4}{(p + 6)(p + 5)} $ The least common multiple of the denominators is: $ (p + 6)(p + 5)$ In order to get the first term over $(p + 6)(p + 5)$ , multiply by $\dfrac{4(p + 5)}{4(p + 5)}$ $ \dfrac{5}{p + 6} \times \dfrac{4(p + 5)}{4(p + 5)} = \dfrac{20(p + 5)}{(p + 6)(p + 5)} $ In order to get the second term over $(p + 6)(p + 5)$ , multiply by $\dfrac{p + 6}{p + 6}$ $ \dfrac{5}{4(p + 5)} \times \dfrac{p + 6}{p + 6} = \dfrac{5(p + 6)}{(p + 6)(p + 5)} $ In order to get the third term over $(p + 6)(p + 5)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{4}{(p + 6)(p + 5)} \times \dfrac{4}{4} = \dfrac{16}{(p + 6)(p + 5)} $ Now we have: $ \dfrac{20(p + 5)}{(p + 6)(p + 5)} - \dfrac{5(p + 6)}{(p + 6)(p + 5)} - \dfrac{16}{(p + 6)(p + 5)} $ $ = \dfrac{ 20(p + 5) - 5(p + 6) - 16} {(p + 6)(p + 5)} $ Expand: $ = \dfrac{20p + 100 - 5p - 30 - 16}{4p^2 + 44p + 120} $ $ = \dfrac{15p + 54}{4p^2 + 44p + 120}$